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[Physics.Soc-Ph] 4 Jun 2020 5 Last Words 7 Recent advances in opinion propagation dynamics: A 2020 Survey Hossein Noorazar∗1 1Washington State University, Pullman, Washington, United States of America Abstract Opinion dynamics have attracted the interest of researchers from different fields. Local interactions among individuals create interesting dynamics for the system as a whole. Such dynamics are important from a variety of perspectives. Group decision making, successful marketing, and constructing networks (in which consensus can be reached or prevented) are a few examples of existing or potential applications. The invention of the Internet has made the opinion fusion faster, unilateral, and on a whole different scale. Spread of fake news, propaganda, and election interferences have made it clear there is an essential need to know more about these dynamics. The emergence of new ideas in the field has accelerated over the last few years. In the first quarter of 2020, at least 50 research papers have emerged, either peer-reviewed and published or on pre-print outlets such as arXiv. In this paper, we summarize these ground-breaking ideas and their fascinating extensions and introduce newly surfaced concepts. Keywords:— Opinion dynamics, social dynamics, social interaction, consensus, polarization Contents 1 Introduction 1 2 Preliminaries 2 3 Milestones 2 3.1 Continuous opinion space models . .2 3.1.1 DeGrootian models . .2 3.1.2 Bounded confidence models . .3 3.2 Discrete opinion space models . .3 3.2.1 Galam model . .3 3.2.2 Sznajd model . .3 3.2.3 Voter model . .4 4 Milestones’ extensions 4 4.1 Stubborn agents . .4 4.2 Biased agents . .4 4.3 Opinion manipulation . .4 4.4 Power evolution . .5 4.5 Repulsive behavior . .5 4.6 Noisy models . .6 4.7 Interrelated topics . .6 4.8 Expressed vs. private opinions . .6 arXiv:2004.05286v2 [physics.soc-ph] 4 Jun 2020 5 Last words 7 6 New questions 7 7 Conclusions 8 8 Acknowledgments 8 1 Introduction ied the effect of group pressure on social dynamics. French in 1956 was among the first researchers to devote attention to opinion dynamics; A Formal Theory of Social Power [2]. Opinion dynamics studies propagation of opinions in a net- In 1964 Abelson [3] proposed a continuous-time model. A work through interactions of its agents. Modeling opinion decade later, in 1974, DeGroot established one of the sim- dynamics goes back a few decades. Asch in 1951 [1] stud- ∗[email protected] 1 plest models [4], which has become one of the most well An agent that does not change its opinion over time is re- known. Two years later Lehrer [5] also developed his model ferred to as a fully-stubborn agent. An agent that weighs its that is identical to that of DeGroot and an extensive discus- initial opinion, i.e., takes its initial opinion into account, in sion by Lehrer and Wagner are given in [6]. Another pioneer all interactions over time is referred to as a partially-stubborn and interesting work is [7] by Latané where he explored the agent. If an agent is not stubborn, it is called a non-stubborn interactions from a psychology angle and reviewed some of agent. In the bounded confidence models, we say an agent the earlier works such as group pressure, immitation, and is closed-minded if its confidence radius is smaller than that effects of newspapers. of others. Opinion dynamics take different shapes depending on the nature of the topic under consideration and the purpose of interactions. For example, the topic could be liking or dis- 3 Milestones liking a certain food such as fish; here, binary opinion dy- namics comes into play [8–10]. Sometimes the opinion can In this section, we present the main models that have in- be represented by continuous variables. For example, the spired a tremendous amount of research. We start with the extent to which one supports a cause. Over the years, a models in which the opinion space is continuous and then few different models for continuous-opinions have been pro- present the models in which the opinion space is discrete. posed [2, 11, 12]. Either as an abstract idea or in real life, one can think of a continuous-opinion space in which one 3.1 Continuous opinion space models must take discrete actions [13, 14]. For instance, in an elec- tion where each agent’s support for candidates falls on the In this section, we present the DeGroot model and its ma- continuous spectrum, each agent must still cast a discrete jor extension known as the Friedkin-Johnsen model. Next, vote. we examine the bounded confidence models in which agents In short, opinion dynamics can be explained as follows. interact with those whose opinions are close enough to that Agents start with an initial opinion. Connected agents in- of their own. teract and update their opinion by a given clear update rule. This process is carried on until a termination criterion is met. 3.1.1 DeGrootian models An example of termination criteria is reaching a steady state in which agents do not change their opinion anymore. We begin with the simplest model, the DeGroot model. In this paper, we briefly introduce the main concepts and Moreover, since the Friedkin-Johnsen extension of the DeG- newly developed ideas of modeling opinion dynamics. Go- root model is well-known and has been studied extensively, ing into details is not the purpose of this survey. In Sec.3 we present it here as well. the major models are presented, then, in Sec.4 basic re- DeGroot model. The DeGroot model [4] is given by sults and certain extensions of the well-known models are reviewed. In Sec.5, we go through some models that have o(t) = Wo(t−1) = W2o(t−2) = ··· = Wto(0) (1) not been studied exhaustively but are interesting and have contributed novel concepts. Finally, the conclusions are pre- where W is a row-stochastic weight matrix and Wt is its t-th sented in Sec.7. power. The model is linear and traceable over time. Classi- cal linear algebra tools are sufficient to analyze this model. It has been very well studied and different extensions of it 2 Preliminaries exist. The DeGroot model is an iterative averaging model. If the network is connected then convergence is equivalent to The network of agents is denoted by G in which N agents consensus (Fig.1). Berger [22] showed the DeGroot model are present. Let A be the adjacency matrix where Aij = 1 will reach consensus if and only if there exists a power t of if agents i and j are connected and Aij = 0 otherwise. The the weight matrix for which Wt has a strictly positive col- row-stochastic influence matrix is denoted by W where Wij umn. Let W be such a matrix; then, the consensus opinion is the level of influence of agent j on agent i; 0 ≤ Wij ≤ 1. ∗ (0) T is given by o = h`A; o i where `W is the left eigenvector Let us define the opinion space to be the set of all pos- of W associated with 1, constrained to h1N ; `Ai = 1. sible opinions denoted by O. Examples of opinion space are Friedkin-Johnsen model. One of the major exten- O = f0; 1g; f1; 2; : : : mg; [0; 1]. The opinion of agent i at sions of the DeGroot model was introduced by Friedkin and (t) time t is denoted by oi . The state of the system at time t Johnsen [23, 24] and is known as the Friedkin-Johnsen (FJ) (t) (t) (t) (t) o o ··· o is denoted by o = 1 2 N . model. Since it has functioned as a ground-breaking model, When the system reaches equilibrium, we say the system we include it here instead of in Sec.4. In the FJ model, the has converged. The convergence state may be consensus, idea of stubborn-agents is added to the DeGroot model: polarization, or fragmentation. Polarization is the state in which there are only two clusters of agents, and fragmenta- o(t+1) = DWo(t) + (I − D)o(0) (2) tion is the state there are more than two clusters. Before moving to the next section, we would like to men- where D = diag([d1; d2; : : : ; dN ]) with entries that spec- tion that there is no convergence on terminology in the liter- ify the susceptibility of individual agents to influence, i.e., ature. For example, consider an agent that does not change (1 − di) is the level of stubbornness of agent i. For a fully- its opinion over time; some papers refer to such an agent as stubborn agent 1 − di=1, for a partially-stubborn agent a leader [15], others as a media [16], some as an stubborn 0 < 1 − di < 1 and for a non-stubborn agent 1 − di = 0. W agent [17] and a few as an inflexible agent [18,19]. A closed- is a row stochastic influence matrix. The convergence and minded agent is referred to someone who does not change its stability of the FJ model are studied in [25]. opinion in [20] and in some papers, a closed-minded agent is an agent whose confidence radius is small compared to These two models are the main two DeGrootian models. other agents [21]. We adhere to the following definitions. We now move on to the bounded confidence models. 2 Figure 1: Evolution of the DeGroot model from initial profile to consensus. 3.1.2 Bounded confidence models Hegselmann-Krause model. The most well-known synchronous version of BCM is given by Hegselmann- In this section, we look at bounded confidence models.
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